LMFDB - Embedded newform 4608.2.k.bg.1153.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1153.3
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.bg.3457.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23607 - 2.23607i) q^{5} -1.74806i q^{7} +O(q^{10})$	$q+(2.23607 - 2.23607i) q^{5} -1.74806i q^{7} +(-4.57649 + 4.57649i) q^{11} +(1.00000 + 1.00000i) q^{13} +2.47214 q^{17} +(4.57649 + 4.57649i) q^{19} -5.65685i q^{23} -5.00000i q^{25} +(-4.23607 - 4.23607i) q^{29} -3.90879 q^{31} +(-3.90879 - 3.90879i) q^{35} +(7.47214 - 7.47214i) q^{37} +2.47214i q^{41} +(1.08036 - 1.08036i) q^{43} +3.94427 q^{49} +(8.23607 - 8.23607i) q^{53} +20.4667i q^{55} +(6.32456 - 6.32456i) q^{59} +(-1.47214 - 1.47214i) q^{61} +4.47214 q^{65} +(8.48528 + 8.48528i) q^{67} -9.15298i q^{71} +2.94427i q^{73} +(8.00000 + 8.00000i) q^{77} +7.40492 q^{79} +(-4.57649 - 4.57649i) q^{83} +(5.52786 - 5.52786i) q^{85} -10.0000i q^{89} +(1.74806 - 1.74806i) q^{91} +20.4667 q^{95} -12.9443 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} - 16 q^{17} - 16 q^{29} + 24 q^{37} - 40 q^{49} + 48 q^{53} + 24 q^{61} + 64 q^{77} + 80 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 16 * q^17 - 16 * q^29 + 24 * q^37 - 40 * q^49 + 48 * q^53 + 24 * q^61 + 64 * q^77 + 80 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.23607	−	2.23607i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$5$	2.23607	−	2.23607i	1.00000	−	1.00000i	1.00000			$0$
$6$		0			0
$7$		−	1.74806i		−	0.660706i	−0.943857	−	0.330353i	$-0.892832\pi$
$7$		−	1.74806i		−	0.660706i	0.943857	−	0.330353i	$-0.107168\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.57649	+	4.57649i	−1.37986	+	1.37986i	−0.535033	+	0.844831i	$0.679701\pi$
$11$	−4.57649	+	4.57649i	−1.37986	+	1.37986i	−0.844831	+	0.535033i	$0.820299\pi$
$12$		0			0
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	0.832050	−	0.554700i	$-0.187167\pi$
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.47214			0.599581			0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214			0.599581			0.299791	+	0.954005i	$0.403083\pi$
$18$		0			0
$19$	4.57649	+	4.57649i	1.04992	+	1.04992i	0.998687	+	0.0512323i	$0.0163149\pi$
$19$	4.57649	+	4.57649i	1.04992	+	1.04992i	0.0512323	+	0.998687i	$0.483685\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.65685i		−	1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$		−	5.65685i		−	1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.23607	−	4.23607i	−0.786618	−	0.786618i	0.194320	−	0.980938i	$-0.437750\pi$
$29$	−4.23607	−	4.23607i	−0.786618	−	0.786618i	−0.980938	+	0.194320i	$0.937750\pi$
$30$		0			0
$31$	−3.90879			−0.702039			−0.351020	−	0.936368i	$-0.614165\pi$
$31$	−3.90879			−0.702039			−0.351020	+	0.936368i	$0.614165\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.90879	−	3.90879i	−0.660706	−	0.660706i
$36$		0			0
$37$	7.47214	−	7.47214i	1.22841	−	1.22841i	0.263847	−	0.964565i	$-0.415009\pi$
$37$	7.47214	−	7.47214i	1.22841	−	1.22841i	0.964565	−	0.263847i	$-0.0849914\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.47214i			0.386083i	0.981191	+	0.193041i	$0.0618352\pi$
$41$			2.47214i			0.386083i	−0.981191	+	0.193041i	$0.938165\pi$
$42$		0			0
$43$	1.08036	−	1.08036i	0.164754	−	0.164754i	−0.619915	−	0.784669i	$-0.712833\pi$
$43$	1.08036	−	1.08036i	0.164754	−	0.164754i	0.784669	+	0.619915i	$0.212833\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	3.94427			0.563467
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.23607	−	8.23607i	1.13131	−	1.13131i	0.141351	−	0.989959i	$-0.454855\pi$
$53$	8.23607	−	8.23607i	1.13131	−	1.13131i	0.989959	−	0.141351i	$-0.0451448\pi$
$54$		0			0
$55$			20.4667i			2.75973i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.32456	−	6.32456i	0.823387	−	0.823387i	−0.163205	−	0.986592i	$-0.552183\pi$
$59$	6.32456	−	6.32456i	0.823387	−	0.823387i	0.986592	+	0.163205i	$0.0521833\pi$
$60$		0			0
$61$	−1.47214	−	1.47214i	−0.188488	−	0.188488i	0.606554	−	0.795042i	$-0.292551\pi$
$61$	−1.47214	−	1.47214i	−0.188488	−	0.188488i	−0.795042	+	0.606554i	$0.792551\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.47214			0.554700
$66$		0			0
$67$	8.48528	+	8.48528i	1.03664	+	1.03664i	0.999303	+	0.0373395i	$0.0118883\pi$
$67$	8.48528	+	8.48528i	1.03664	+	1.03664i	0.0373395	+	0.999303i	$0.488112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.15298i		−	1.08626i	−0.839649	−	0.543130i	$-0.817239\pi$
$71$		−	9.15298i		−	1.08626i	0.839649	−	0.543130i	$-0.182761\pi$
$72$		0			0
$73$			2.94427i			0.344601i	0.985044	+	0.172300i	$0.0551200\pi$
$73$			2.94427i			0.344601i	−0.985044	+	0.172300i	$0.944880\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.00000	+	8.00000i	0.911685	+	0.911685i
$78$		0			0
$79$	7.40492			0.833118			0.416559	−	0.909109i	$-0.363236\pi$
$79$	7.40492			0.833118			0.416559	+	0.909109i	$0.363236\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.57649	−	4.57649i	−0.502335	−	0.502335i	0.409828	−	0.912163i	$-0.365589\pi$
$83$	−4.57649	−	4.57649i	−0.502335	−	0.502335i	−0.912163	+	0.409828i	$0.865589\pi$
$84$		0			0
$85$	5.52786	−	5.52786i	0.599581	−	0.599581i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.0000i		−	1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$		−	10.0000i		−	1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$		0			0
$91$	1.74806	−	1.74806i	0.183247	−	0.183247i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	20.4667			2.09984
$96$		0			0
$97$	−12.9443			−1.31429			−0.657146	−	0.753763i	$-0.728236\pi$
$97$	−12.9443			−1.31429			−0.657146	+	0.753763i	$0.728236\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.236068	−	0.236068i	0.0234896	−	0.0234896i	−0.695264	−	0.718754i	$-0.744713\pi$
$101$	0.236068	−	0.236068i	0.0234896	−	0.0234896i	0.718754	+	0.695264i	$0.244713\pi$
$102$		0			0
$103$			1.74806i			0.172242i	0.996285	+	0.0861209i	$0.0274471\pi$
$103$			1.74806i			0.172242i	−0.996285	+	0.0861209i	$0.972553\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.667701	−	0.667701i	0.0645491	−	0.0645491i	−0.674095	−	0.738644i	$-0.735466\pi$
$107$	0.667701	−	0.667701i	0.0645491	−	0.0645491i	0.738644	+	0.674095i	$0.235466\pi$
$108$		0			0
$109$	11.9443	+	11.9443i	1.14405	+	1.14405i	0.987701	+	0.156352i	$0.0499736\pi$
$109$	11.9443	+	11.9443i	1.14405	+	1.14405i	0.156352	+	0.987701i	$0.450026\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	−12.6491	−	12.6491i	−1.17954	−	1.17954i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.32145i		−	0.396147i
$120$		0			0
$121$		−	30.8885i		−	2.80805i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−10.9010			−0.967311			−0.483656	−	0.875258i	$-0.660691\pi$
$127$	−10.9010			−0.967311			−0.483656	+	0.875258i	$0.660691\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.9814	−	11.9814i	−1.04682	−	1.04682i	−0.998849	−	0.0479719i	$-0.984724\pi$
$131$	−11.9814	−	11.9814i	−1.04682	−	1.04682i	−0.0479719	−	0.998849i	$-0.515276\pi$
$132$		0			0
$133$	8.00000	−	8.00000i	0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.47214i		−	0.211209i	−0.994408	−	0.105604i	$-0.966322\pi$
$137$		−	2.47214i		−	0.211209i	0.994408	−	0.105604i	$-0.0336777\pi$
$138$		0			0
$139$	11.9814	−	11.9814i	1.01625	−	1.01625i	0.0163841	−	0.999866i	$-0.494785\pi$
$139$	11.9814	−	11.9814i	1.01625	−	1.01625i	0.999866	−	0.0163841i	$-0.00521546\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.15298			−0.765411
$144$		0			0
$145$	−18.9443			−1.57324
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.70820	+	2.70820i	−0.221865	+	0.221865i	−0.809283	−	0.587419i	$-0.800144\pi$
$149$	−2.70820	+	2.70820i	−0.221865	+	0.221865i	0.587419	+	0.809283i	$0.300144\pi$
$150$		0			0
$151$		−	20.0540i		−	1.63197i	−0.578070	−	0.815987i	$-0.696194\pi$
$151$		−	20.0540i		−	1.63197i	0.578070	−	0.815987i	$-0.303806\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.74032	+	8.74032i	−0.702039	+	0.702039i
$156$		0			0
$157$	−9.47214	−	9.47214i	−0.755959	−	0.755959i	0.219626	−	0.975584i	$-0.429516\pi$
$157$	−9.47214	−	9.47214i	−0.755959	−	0.755959i	−0.975584	+	0.219626i	$0.929516\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.88854			−0.779326
$162$		0			0
$163$	10.2333	+	10.2333i	0.801537	+	0.801537i	0.983336	−	0.181798i	$-0.0581918\pi$
$163$	10.2333	+	10.2333i	0.801537	+	0.801537i	−0.181798	+	0.983336i	$0.558192\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.4667i			1.58376i	0.610677	+	0.791880i	$0.290898\pi$
$167$			20.4667i			1.58376i	−0.610677	+	0.791880i	$0.709102\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.70820	+	6.70820i	0.510015	+	0.510015i	0.914531	−	0.404516i	$-0.132560\pi$
$173$	6.70820	+	6.70820i	0.510015	+	0.510015i	−0.404516	+	0.914531i	$0.632560\pi$
$174$		0			0
$175$	−8.74032			−0.660706
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.48528	+	8.48528i	0.634220	+	0.634220i	0.949124	−	0.314904i	$-0.101972\pi$
$179$	8.48528	+	8.48528i	0.634220	+	0.634220i	−0.314904	+	0.949124i	$0.601972\pi$
$180$		0			0
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	−0.168140	−	0.985763i	$-0.553776\pi$
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	0.985763	+	0.168140i	$0.0537759\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	33.4164i		−	2.45682i
$186$		0			0
$187$	−11.3137	+	11.3137i	−0.827340	+	0.827340i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.8098			−1.07160			−0.535801	−	0.844344i	$-0.679990\pi$
$191$	−14.8098			−1.07160			−0.535801	+	0.844344i	$0.679990\pi$
$192$		0			0
$193$	4.94427			0.355896			0.177948	−	0.984040i	$-0.443054\pi$
$193$	4.94427			0.355896			0.177948	+	0.984040i	$0.443054\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.18034	+	5.18034i	−0.369084	+	0.369084i	−0.867143	−	0.498059i	$-0.834046\pi$
$197$	−5.18034	+	5.18034i	−0.369084	+	0.369084i	0.498059	+	0.867143i	$0.334046\pi$
$198$		0			0
$199$			5.24419i			0.371751i	0.982573	+	0.185875i	$0.0595121\pi$
$199$			5.24419i			0.371751i	−0.982573	+	0.185875i	$0.940488\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.40492	+	7.40492i	−0.519723	+	0.519723i
$204$		0			0
$205$	5.52786	+	5.52786i	0.386083	+	0.386083i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−41.8885			−2.89749
$210$		0			0
$211$	−4.98915	−	4.98915i	−0.343467	−	0.343467i	0.514202	−	0.857669i	$-0.328088\pi$
$211$	−4.98915	−	4.98915i	−0.343467	−	0.343467i	−0.857669	+	0.514202i	$0.828088\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.83153i		−	0.329508i
$216$		0			0
$217$			6.83282i			0.463842i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.47214	+	2.47214i	0.166294	+	0.166294i
$222$		0			0
$223$	−3.90879			−0.261752			−0.130876	−	0.991399i	$-0.541779\pi$
$223$	−3.90879			−0.261752			−0.130876	+	0.991399i	$0.541779\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.2333	+	10.2333i	0.679211	+	0.679211i	0.959822	−	0.280611i	$-0.0905370\pi$
$227$	10.2333	+	10.2333i	0.679211	+	0.679211i	−0.280611	+	0.959822i	$0.590537\pi$
$228$		0			0
$229$	−9.94427	+	9.94427i	−0.657136	+	0.657136i	−0.954701	−	0.297565i	$-0.903825\pi$
$229$	−9.94427	+	9.94427i	−0.657136	+	0.657136i	0.297565	+	0.954701i	$0.403825\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.1235			−1.68979			−0.844896	−	0.534931i	$-0.820338\pi$
$239$	−26.1235			−1.68979			−0.844896	+	0.534931i	$0.820338\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.81966	−	8.81966i	0.563467	−	0.563467i
$246$		0			0
$247$			9.15298i			0.582390i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.07262	−	8.07262i	0.509539	−	0.509539i	−0.404846	−	0.914385i	$-0.632675\pi$
$251$	8.07262	−	8.07262i	0.509539	−	0.509539i	0.914385	+	0.404846i	$0.132675\pi$
$252$		0			0
$253$	25.8885	+	25.8885i	1.62760	+	1.62760i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$		0			0
$259$	−13.0618	−	13.0618i	−0.811619	−	0.811619i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			12.6491i			0.779978i	0.920820	+	0.389989i	$0.127521\pi$
$263$			12.6491i			0.779978i	−0.920820	+	0.389989i	$0.872479\pi$
$264$		0			0
$265$		−	36.8328i		−	2.26262i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.76393	+	3.76393i	0.229491	+	0.229491i	0.812480	−	0.582989i	$-0.198117\pi$
$269$	3.76393	+	3.76393i	0.229491	+	0.229491i	−0.582989	+	0.812480i	$0.698117\pi$
$270$		0			0
$271$	10.9010			0.662191			0.331096	−	0.943597i	$-0.392582\pi$
$271$	10.9010			0.662191			0.331096	+	0.943597i	$0.392582\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.8825	+	22.8825i	1.37986	+	1.37986i
$276$		0			0
$277$	13.0000	−	13.0000i	0.781094	−	0.781094i	−0.198921	−	0.980015i	$-0.563744\pi$
$277$	13.0000	−	13.0000i	0.781094	−	0.781094i	0.980015	+	0.198921i	$0.0637438\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.88854i			0.231971i	0.993251	+	0.115986i	$0.0370026\pi$
$281$			3.88854i			0.231971i	−0.993251	+	0.115986i	$0.962997\pi$
$282$		0			0
$283$	−18.9737	+	18.9737i	−1.12787	+	1.12787i	−0.137344	+	0.990523i	$0.543857\pi$
$283$	−18.9737	+	18.9737i	−1.12787	+	1.12787i	−0.990523	+	0.137344i	$0.956143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.32145			0.255087
$288$		0			0
$289$	−10.8885			−0.640503
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.2361	+	16.2361i	−0.948521	+	0.948521i	−0.998738	−	0.0502171i	$-0.984009\pi$
$293$	−16.2361	+	16.2361i	−0.948521	+	0.948521i	0.0502171	+	0.998738i	$0.484009\pi$
$294$		0			0
$295$		−	28.2843i		−	1.64677i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.65685	−	5.65685i	0.327144	−	0.327144i
$300$		0			0
$301$	−1.88854	−	1.88854i	−0.108854	−	0.108854i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.58359			−0.376975
$306$		0			0
$307$	4.16383	+	4.16383i	0.237642	+	0.237642i	0.815873	−	0.578231i	$-0.196257\pi$
$307$	4.16383	+	4.16383i	0.237642	+	0.237642i	−0.578231	+	0.815873i	$0.696257\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	19.6414i		−	1.11376i	−0.830593	−	0.556880i	$-0.811998\pi$
$311$		−	19.6414i		−	1.11376i	0.830593	−	0.556880i	$-0.188002\pi$
$312$		0			0
$313$		−	20.9443i		−	1.18384i	−0.805997	−	0.591920i	$-0.798370\pi$
$313$		−	20.9443i		−	1.18384i	0.805997	−	0.591920i	$-0.201630\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.2361	−	12.2361i	−0.687246	−	0.687246i	0.274376	−	0.961622i	$-0.411529\pi$
$317$	−12.2361	−	12.2361i	−0.687246	−	0.687246i	−0.961622	+	0.274376i	$0.911529\pi$
$318$		0			0
$319$	38.7727			2.17085
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.3137	+	11.3137i	0.629512	+	0.629512i
$324$		0			0
$325$	5.00000	−	5.00000i	0.277350	−	0.277350i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.9814	+	11.9814i	−0.658558	+	0.658558i	−0.955039	−	0.296481i	$-0.904187\pi$
$331$	−11.9814	+	11.9814i	−0.658558	+	0.658558i	0.296481	+	0.955039i	$0.404187\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	37.9473			2.07328
$336$		0			0
$337$	−14.9443			−0.814066			−0.407033	−	0.913413i	$-0.633437\pi$
$337$	−14.9443			−0.814066			−0.407033	+	0.913413i	$0.633437\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.8885	−	17.8885i	0.968719	−	0.968719i
$342$		0			0
$343$		−	19.1313i		−	1.03299i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.8902	−	15.8902i	0.853031	−	0.853031i	−0.137475	−	0.990505i	$-0.543899\pi$
$347$	15.8902	−	15.8902i	0.853031	−	0.853031i	0.990505	+	0.137475i	$0.0438986\pi$
$348$		0			0
$349$	−6.52786	−	6.52786i	−0.349429	−	0.349429i	0.510468	−	0.859897i	$-0.329472\pi$
$349$	−6.52786	−	6.52786i	−0.349429	−	0.349429i	−0.859897	+	0.510468i	$0.829472\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.8885			0.632763			0.316382	−	0.948632i	$-0.397532\pi$
$353$	11.8885			0.632763			0.316382	+	0.948632i	$0.397532\pi$
$354$		0			0
$355$	−20.4667	−	20.4667i	−1.08626	−	1.08626i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			13.4744i			0.711153i	0.934647	+	0.355577i	$0.115716\pi$
$359$			13.4744i			0.711153i	−0.934647	+	0.355577i	$0.884284\pi$
$360$		0			0
$361$			22.8885i			1.20466i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.58359	+	6.58359i	0.344601	+	0.344601i
$366$		0			0
$367$	17.8933			0.934023			0.467011	−	0.884251i	$-0.345331\pi$
$367$	17.8933			0.934023			0.467011	+	0.884251i	$0.345331\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.3972	−	14.3972i	−0.747464	−	0.747464i
$372$		0			0
$373$	15.4721	−	15.4721i	0.801117	−	0.801117i	−0.182153	−	0.983270i	$-0.558307\pi$
$373$	15.4721	−	15.4721i	0.801117	−	0.801117i	0.983270	+	0.182153i	$0.0583067\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.47214i		−	0.436337i
$378$		0			0
$379$	19.3863	−	19.3863i	0.995809	−	0.995809i	−0.00418248	−	0.999991i	$-0.501331\pi$
$379$	19.3863	−	19.3863i	0.995809	−	0.995809i	0.999991	+	0.00418248i	$0.00133133\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−19.1313			−0.977563			−0.488782	−	0.872406i	$-0.662559\pi$
$383$	−19.1313			−0.977563			−0.488782	+	0.872406i	$0.662559\pi$
$384$		0			0
$385$	35.7771			1.82337
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.6525	−	23.6525i	1.19923	−	1.19923i	0.224830	−	0.974398i	$-0.427817\pi$
$389$	23.6525	−	23.6525i	1.19923	−	1.19923i	0.974398	−	0.224830i	$-0.0721827\pi$
$390$		0			0
$391$		−	13.9845i		−	0.707227i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.5579	−	16.5579i	0.833118	−	0.833118i
$396$		0			0
$397$	6.41641	+	6.41641i	0.322030	+	0.322030i	0.849546	−	0.527515i	$-0.176876\pi$
$397$	6.41641	+	6.41641i	0.322030	+	0.322030i	−0.527515	+	0.849546i	$0.676876\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.5279			0.675549			0.337775	−	0.941227i	$-0.390326\pi$
$401$	13.5279			0.675549			0.337775	+	0.941227i	$0.390326\pi$
$402$		0			0
$403$	−3.90879	−	3.90879i	−0.194711	−	0.194711i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			68.3923i			3.39008i
$408$		0			0
$409$			12.9443i			0.640053i	0.947409	+	0.320027i	$0.103692\pi$
$409$			12.9443i			0.640053i	−0.947409	+	0.320027i	$0.896308\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11.0557	−	11.0557i	−0.544017	−	0.544017i
$414$		0			0
$415$	−20.4667			−1.00467
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.08036	+	1.08036i	0.0527792	+	0.0527792i	0.733004	−	0.680225i	$-0.238118\pi$
$419$	1.08036	+	1.08036i	0.0527792	+	0.0527792i	−0.680225	+	0.733004i	$0.738118\pi$
$420$		0			0
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	−0.818373	−	0.574688i	$-0.805124\pi$
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	0.574688	+	0.818373i	$0.305124\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	12.3607i		−	0.599581i
$426$		0			0
$427$	−2.57339	+	2.57339i	−0.124535	+	0.124535i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.3060			0.881767			0.440884	−	0.897564i	$-0.354665\pi$
$431$	18.3060			0.881767			0.440884	+	0.897564i	$0.354665\pi$
$432$		0			0
$433$	−24.0000			−1.15337			−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	25.8885	−	25.8885i	1.23842	−	1.23842i
$438$		0			0
$439$			6.06952i			0.289682i	0.989455	+	0.144841i	$0.0462671\pi$
$439$			6.06952i			0.289682i	−0.989455	+	0.144841i	$0.953733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.07262	+	8.07262i	−0.383542	+	0.383542i	−0.872376	−	0.488835i	$-0.837422\pi$
$443$	−8.07262	+	8.07262i	−0.383542	+	0.383542i	0.488835	+	0.872376i	$0.337422\pi$
$444$		0			0
$445$	−22.3607	−	22.3607i	−1.06000	−	1.06000i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.4164			1.10509			0.552544	−	0.833484i	$-0.313657\pi$
$449$	23.4164			1.10509			0.552544	+	0.833484i	$0.313657\pi$
$450$		0			0
$451$	−11.3137	−	11.3137i	−0.532742	−	0.532742i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	7.81758i		−	0.366494i
$456$		0			0
$457$			3.05573i			0.142941i	0.997443	+	0.0714705i	$0.0227692\pi$
$457$			3.05573i			0.142941i	−0.997443	+	0.0714705i	$0.977231\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.23607	+	6.23607i	0.290443	+	0.290443i	0.837255	−	0.546812i	$-0.184159\pi$
$461$	6.23607	+	6.23607i	0.290443	+	0.290443i	−0.546812	+	0.837255i	$0.684159\pi$
$462$		0			0
$463$	−7.40492			−0.344136			−0.172068	−	0.985085i	$-0.555045\pi$
$463$	−7.40492			−0.344136			−0.172068	+	0.985085i	$0.555045\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.24109	−	3.24109i	−0.149980	−	0.149980i	0.628129	−	0.778109i	$-0.283821\pi$
$467$	−3.24109	−	3.24109i	−0.149980	−	0.149980i	−0.778109	+	0.628129i	$0.783821\pi$
$468$		0			0
$469$	14.8328	−	14.8328i	0.684916	−	0.684916i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			9.88854i			0.454676i
$474$		0			0
$475$	22.8825	−	22.8825i	1.04992	−	1.04992i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.49613			0.159742			0.0798711	−	0.996805i	$-0.474549\pi$
$479$	3.49613			0.159742			0.0798711	+	0.996805i	$0.474549\pi$
$480$		0			0
$481$	14.9443			0.681400
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−28.9443	+	28.9443i	−1.31429	+	1.31429i
$486$		0			0
$487$		−	16.5579i		−	0.750310i	−0.926962	−	0.375155i	$-0.877589\pi$
$487$		−	16.5579i		−	0.750310i	0.926962	−	0.375155i	$-0.122411\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.4775	+	15.4775i	−0.698492	+	0.698492i	−0.964085	−	0.265594i	$-0.914432\pi$
$491$	−15.4775	+	15.4775i	−0.698492	+	0.698492i	0.265594	+	0.964085i	$0.414432\pi$
$492$		0			0
$493$	−10.4721	−	10.4721i	−0.471641	−	0.471641i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000			−0.717698
$498$		0			0
$499$	17.6383	+	17.6383i	0.789597	+	0.789597i	0.981428	−	0.191831i	$-0.0614425\pi$
$499$	17.6383	+	17.6383i	0.789597	+	0.789597i	−0.191831	+	0.981428i	$0.561443\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			4.83153i			0.215427i	0.994182	+	0.107714i	$0.0343530\pi$
$503$			4.83153i			0.215427i	−0.994182	+	0.107714i	$0.965647\pi$
$504$		0			0
$505$		−	1.05573i		−	0.0469793i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	28.2361	+	28.2361i	1.25154	+	1.25154i	0.955029	+	0.296513i	$0.0958238\pi$
$509$	28.2361	+	28.2361i	1.25154	+	1.25154i	0.296513	+	0.955029i	$0.404176\pi$
$510$		0			0
$511$	5.14678			0.227680
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.90879	+	3.90879i	0.172242	+	0.172242i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			28.3607i			1.24250i	0.783611	+	0.621252i	$0.213376\pi$
$521$			28.3607i			1.24250i	−0.783611	+	0.621252i	$0.786624\pi$
$522$		0			0
$523$	−11.5687	+	11.5687i	−0.505866	+	0.505866i	−0.913255	−	0.407389i	$-0.866439\pi$
$523$	−11.5687	+	11.5687i	−0.505866	+	0.505866i	0.407389	+	0.913255i	$0.366439\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.66306			−0.420930
$528$		0			0
$529$	−9.00000			−0.391304
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.47214	+	2.47214i	−0.107080	+	0.107080i
$534$		0			0
$535$		−	2.98605i		−	0.129098i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.0509	+	18.0509i	−0.777508	+	0.777508i
$540$		0			0
$541$	18.8885	+	18.8885i	0.812082	+	0.812082i	0.984946	−	0.172864i	$-0.0553020\pi$
$541$	18.8885	+	18.8885i	0.812082	+	0.812082i	−0.172864	+	0.984946i	$0.555302\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	53.4164			2.28811
$546$		0			0
$547$	22.8825	+	22.8825i	0.978383	+	0.978383i	0.999771	−	0.0213881i	$-0.00680856\pi$
$547$	22.8825	+	22.8825i	0.978383	+	0.978383i	−0.0213881	+	0.999771i	$0.506809\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	38.7727i		−	1.65177i
$552$		0			0
$553$		−	12.9443i		−	0.550446i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.1803	−	11.1803i	−0.473726	−	0.473726i	0.429392	−	0.903118i	$-0.358728\pi$
$557$	−11.1803	−	11.1803i	−0.473726	−	0.473726i	−0.903118	+	0.429392i	$0.858728\pi$
$558$		0			0
$559$	2.16073			0.0913890
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.08036	+	1.08036i	0.0455319	+	0.0455319i	0.729506	−	0.683974i	$-0.239750\pi$
$563$	1.08036	+	1.08036i	0.0455319	+	0.0455319i	−0.683974	+	0.729506i	$0.739750\pi$
$564$		0			0
$565$	−4.47214	+	4.47214i	−0.188144	+	0.188144i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7.41641i			0.310912i	0.987843	+	0.155456i	$0.0496847\pi$
$569$			7.41641i			0.310912i	−0.987843	+	0.155456i	$0.950315\pi$
$570$		0			0
$571$	6.32456	−	6.32456i	0.264674	−	0.264674i	−0.562276	−	0.826950i	$-0.690074\pi$
$571$	6.32456	−	6.32456i	0.264674	−	0.264674i	0.826950	+	0.562276i	$0.190074\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.2843			−1.17954
$576$		0			0
$577$	26.9443			1.12170			0.560852	−	0.827916i	$-0.310474\pi$
$577$	26.9443			1.12170			0.560852	+	0.827916i	$0.310474\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.00000	+	8.00000i	−0.331896	+	0.331896i
$582$		0			0
$583$			75.3846i			3.12211i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.2951	−	23.2951i	0.961492	−	0.961492i	−0.0377932	−	0.999286i	$-0.512033\pi$
$587$	23.2951	−	23.2951i	0.961492	−	0.961492i	0.999286	+	0.0377932i	$0.0120328\pi$
$588$		0			0
$589$	−17.8885	−	17.8885i	−0.737085	−	0.737085i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.88854			0.323944			0.161972	−	0.986795i	$-0.448215\pi$
$593$	7.88854			0.323944			0.161972	+	0.986795i	$0.448215\pi$
$594$		0			0
$595$	−9.66306	−	9.66306i	−0.396147	−	0.396147i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			5.65685i			0.231133i	0.993300	+	0.115566i	$0.0368683\pi$
$599$			5.65685i			0.231133i	−0.993300	+	0.115566i	$0.963132\pi$
$600$		0			0
$601$			6.94427i			0.283263i	0.989919	+	0.141631i	$0.0452348\pi$
$601$			6.94427i			0.283263i	−0.989919	+	0.141631i	$0.954765\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−69.0689	−	69.0689i	−2.80805	−	2.80805i
$606$		0			0
$607$	−44.8422			−1.82009			−0.910044	−	0.414512i	$-0.863952\pi$
$607$	−44.8422			−1.82009			−0.910044	+	0.414512i	$0.863952\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7.47214	+	7.47214i	−0.301797	+	0.301797i	−0.841716	−	0.539920i	$-0.818455\pi$
$613$	−7.47214	+	7.47214i	−0.301797	+	0.301797i	0.539920	+	0.841716i	$0.318455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.8885i			0.639649i	0.947477	+	0.319824i	$0.103624\pi$
$617$			15.8885i			0.639649i	−0.947477	+	0.319824i	$0.896376\pi$
$618$		0			0
$619$	−4.16383	+	4.16383i	−0.167358	+	0.167358i	−0.785817	−	0.618459i	$-0.787757\pi$
$619$	−4.16383	+	4.16383i	−0.167358	+	0.167358i	0.618459	+	0.785817i	$0.287757\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−17.4806			−0.700347
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.4721	−	18.4721i	0.736532	−	0.736532i
$630$		0			0
$631$		−	27.0463i		−	1.07670i	−0.842723	−	0.538348i	$-0.819049\pi$
$631$		−	27.0463i		−	1.07670i	0.842723	−	0.538348i	$-0.180951\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.3755	+	24.3755i	−0.967311	+	0.967311i
$636$		0			0
$637$	3.94427	+	3.94427i	0.156278	+	0.156278i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.58359			−0.339032			−0.169516	−	0.985527i	$-0.554220\pi$
$641$	−8.58359			−0.339032			−0.169516	+	0.985527i	$0.554220\pi$
$642$		0			0
$643$	−5.91189	−	5.91189i	−0.233142	−	0.233142i	0.580861	−	0.814003i	$-0.302716\pi$
$643$	−5.91189	−	5.91189i	−0.233142	−	0.233142i	−0.814003	+	0.580861i	$0.802716\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	42.2688i		−	1.66176i	−0.556454	−	0.830879i	$-0.687838\pi$
$647$		−	42.2688i		−	1.66176i	0.556454	−	0.830879i	$-0.312162\pi$
$648$		0			0
$649$			57.8885i			2.27232i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	27.1803	+	27.1803i	1.06365	+	1.06365i	0.997832	+	0.0658172i	$0.0209654\pi$
$653$	27.1803	+	27.1803i	1.06365	+	1.06365i	0.0658172	+	0.997832i	$0.479035\pi$
$654$		0			0
$655$	−53.5825			−2.09364
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.82843	−	2.82843i	−0.110180	−	0.110180i	0.649868	−	0.760047i	$-0.274824\pi$
$659$	−2.82843	−	2.82843i	−0.110180	−	0.110180i	−0.760047	+	0.649868i	$0.774824\pi$
$660$		0			0
$661$	−28.4164	+	28.4164i	−1.10527	+	1.10527i	−0.111506	+	0.993764i	$0.535568\pi$
$661$	−28.4164	+	28.4164i	−1.10527	+	1.10527i	−0.993764	+	0.111506i	$0.964432\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	35.7771i		−	1.38738i
$666$		0			0
$667$	−23.9628	+	23.9628i	−0.927844	+	0.927844i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.4744			0.520175
$672$		0			0
$673$	−17.8885			−0.689553			−0.344776	−	0.938685i	$-0.612045\pi$
$673$	−17.8885			−0.689553			−0.344776	+	0.938685i	$0.612045\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.347524	+	0.347524i	−0.0133564	+	0.0133564i	−0.713753	−	0.700397i	$-0.753006\pi$
$677$	−0.347524	+	0.347524i	−0.0133564	+	0.0133564i	0.700397	+	0.713753i	$0.253006\pi$
$678$		0			0
$679$			22.6274i			0.868361i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.41577	+	2.41577i	−0.0924367	+	0.0924367i	−0.751813	−	0.659376i	$-0.770820\pi$
$683$	−2.41577	+	2.41577i	−0.0924367	+	0.0924367i	0.659376	+	0.751813i	$0.270820\pi$
$684$		0			0
$685$	−5.52786	−	5.52786i	−0.211209	−	0.211209i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.4721			0.627538
$690$		0			0
$691$	18.0509	+	18.0509i	0.686690	+	0.686690i	0.961499	−	0.274809i	$-0.0886147\pi$
$691$	18.0509	+	18.0509i	0.686690	+	0.686690i	−0.274809	+	0.961499i	$0.588615\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	53.5825i		−	2.03250i
$696$		0			0
$697$			6.11146i			0.231488i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	25.1803	+	25.1803i	0.951048	+	0.951048i	0.998857	−	0.0478082i	$-0.0152236\pi$
$701$	25.1803	+	25.1803i	0.951048	+	0.951048i	−0.0478082	+	0.998857i	$0.515224\pi$
$702$		0			0
$703$	68.3923			2.57947
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.412662	−	0.412662i	−0.0155198	−	0.0155198i
$708$		0			0
$709$	−23.9443	+	23.9443i	−0.899246	+	0.899246i	−0.995369	−	0.0961234i	$-0.969356\pi$
$709$	−23.9443	+	23.9443i	−0.899246	+	0.899246i	0.0961234	+	0.995369i	$0.469356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			22.1115i			0.828081i
$714$		0			0
$715$	−20.4667	+	20.4667i	−0.765411	+	0.765411i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.8021			−0.813081			−0.406540	−	0.913633i	$-0.633265\pi$
$719$	−21.8021			−0.813081			−0.406540	+	0.913633i	$0.633265\pi$
$720$		0			0
$721$	3.05573			0.113801
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.1803	+	21.1803i	−0.786618	+	0.786618i
$726$		0			0
$727$			20.0540i			0.743763i	0.928280	+	0.371881i	$0.121287\pi$
$727$			20.0540i			0.743763i	−0.928280	+	0.371881i	$0.878713\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.67080	−	2.67080i	0.0987833	−	0.0987833i
$732$		0			0
$733$	−21.9443	−	21.9443i	−0.810530	−	0.810530i	0.174183	−	0.984713i	$-0.444272\pi$
$733$	−21.9443	−	21.9443i	−0.810530	−	0.810530i	−0.984713	+	0.174183i	$0.944272\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−77.6656			−2.86085
$738$		0			0
$739$	−23.2951	−	23.2951i	−0.856925	−	0.856925i	0.134050	−	0.990975i	$-0.457202\pi$
$739$	−23.2951	−	23.2951i	−0.856925	−	0.856925i	−0.990975	+	0.134050i	$0.957202\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.16073i		−	0.0792694i	−0.999214	−	0.0396347i	$-0.987381\pi$
$743$		−	2.16073i		−	0.0792694i	0.999214	−	0.0396347i	$-0.0126194\pi$
$744$		0			0
$745$			12.1115i			0.443729i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.16718	−	1.16718i	−0.0426480	−	0.0426480i
$750$		0			0
$751$	−7.40492			−0.270209			−0.135105	−	0.990831i	$-0.543137\pi$
$751$	−7.40492			−0.270209			−0.135105	+	0.990831i	$0.543137\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−44.8422	−	44.8422i	−1.63197	−	1.63197i
$756$		0			0
$757$	−9.94427	+	9.94427i	−0.361431	+	0.361431i	−0.864339	−	0.502909i	$-0.832263\pi$
$757$	−9.94427	+	9.94427i	−0.361431	+	0.361431i	0.502909	+	0.864339i	$0.332263\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	7.41641i		−	0.268845i	−0.990924	−	0.134422i	$-0.957082\pi$
$761$		−	7.41641i		−	0.268845i	0.990924	−	0.134422i	$-0.0429179\pi$
$762$		0			0
$763$	20.8794	−	20.8794i	0.755883	−	0.755883i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.6491			0.456733
$768$		0			0
$769$	−30.8328			−1.11186			−0.555930	−	0.831229i	$-0.687638\pi$
$769$	−30.8328			−1.11186			−0.555930	+	0.831229i	$0.687638\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.70820	+	4.70820i	−0.169342	+	0.169342i	−0.786690	−	0.617348i	$-0.788207\pi$
$773$	−4.70820	+	4.70820i	−0.169342	+	0.169342i	0.617348	+	0.786690i	$0.288207\pi$
$774$		0			0
$775$			19.5440i			0.702039i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.3137	+	11.3137i	−0.405356	+	0.405356i
$780$		0			0
$781$	41.8885	+	41.8885i	1.49889	+	1.49889i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−42.3607			−1.51192
$786$		0			0
$787$	10.2333	+	10.2333i	0.364779	+	0.364779i	0.865569	−	0.500790i	$-0.166957\pi$
$787$	10.2333	+	10.2333i	0.364779	+	0.364779i	−0.500790	+	0.865569i	$0.666957\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			3.49613i			0.124308i
$792$		0			0
$793$		−	2.94427i		−	0.104554i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.65248	+	3.65248i	0.129377	+	0.129377i	0.768830	−	0.639453i	$-0.220839\pi$
$797$	3.65248	+	3.65248i	0.129377	+	0.129377i	−0.639453	+	0.768830i	$0.720839\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.4744	−	13.4744i	−0.475502	−	0.475502i
$804$		0			0
$805$	−22.1115	+	22.1115i	−0.779326	+	0.779326i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			23.4164i			0.823277i	0.911347	+	0.411639i	$0.135043\pi$
$809$			23.4164i			0.823277i	−0.911347	+	0.411639i	$0.864957\pi$
$810$		0			0
$811$	20.7217	−	20.7217i	0.727638	−	0.727638i	−0.242510	−	0.970149i	$-0.577971\pi$
$811$	20.7217	−	20.7217i	0.727638	−	0.727638i	0.970149	+	0.242510i	$0.0779709\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	45.7649			1.60307
$816$		0			0
$817$	9.88854			0.345956
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−35.5410	+	35.5410i	−1.24039	+	1.24039i	−0.280551	+	0.959839i	$0.590517\pi$
$821$	−35.5410	+	35.5410i	−1.24039	+	1.24039i	−0.959839	+	0.280551i	$0.909483\pi$
$822$		0			0
$823$		−	1.74806i		−	0.0609337i	−0.999536	−	0.0304668i	$-0.990301\pi$
$823$		−	1.74806i		−	0.0609337i	0.999536	−	0.0304668i	$-0.00969940\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.8067	−	12.8067i	0.445334	−	0.445334i	−0.448466	−	0.893800i	$-0.648030\pi$
$827$	12.8067	−	12.8067i	0.445334	−	0.445334i	0.893800	+	0.448466i	$0.148030\pi$
$828$		0			0
$829$	−3.94427	−	3.94427i	−0.136990	−	0.136990i	0.635286	−	0.772277i	$-0.280882\pi$
$829$	−3.94427	−	3.94427i	−0.136990	−	0.136990i	−0.772277	+	0.635286i	$0.780882\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.75078			0.337844
$834$		0			0
$835$	45.7649	+	45.7649i	1.58376	+	1.58376i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	6.48218i		−	0.223790i	−0.993720	−	0.111895i	$-0.964308\pi$
$839$		−	6.48218i		−	0.223790i	0.993720	−	0.111895i	$-0.0356920\pi$
$840$		0			0
$841$			6.88854i			0.237536i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.5967	−	24.5967i	−0.846154	−	0.846154i
$846$		0			0
$847$	−53.9952			−1.85530
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−42.2688	−	42.2688i	−1.44896	−	1.44896i
$852$		0			0
$853$	−21.4721	+	21.4721i	−0.735192	+	0.735192i	−0.971643	−	0.236451i	$-0.924016\pi$
$853$	−21.4721	+	21.4721i	−0.735192	+	0.735192i	0.236451	+	0.971643i	$0.424016\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			55.4164i			1.89299i	0.322721	+	0.946494i	$0.395403\pi$
$857$			55.4164i			1.89299i	−0.322721	+	0.946494i	$0.604597\pi$
$858$		0			0
$859$	38.5176	−	38.5176i	1.31420	−	1.31420i	0.395918	−	0.918286i	$-0.370426\pi$
$859$	38.5176	−	38.5176i	1.31420	−	1.31420i	0.918286	−	0.395918i	$-0.129574\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.7943			0.980171			0.490086	−	0.871674i	$-0.336966\pi$
$863$	28.7943			0.980171			0.490086	+	0.871674i	$0.336966\pi$
$864$		0			0
$865$	30.0000			1.02003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−33.8885	+	33.8885i	−1.14959	+	1.14959i
$870$		0			0
$871$			16.9706i			0.575026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.3607	+	19.3607i	0.653764	+	0.653764i	0.953897	−	0.300134i	$-0.0970312\pi$
$877$	19.3607	+	19.3607i	0.653764	+	0.653764i	−0.300134	+	0.953897i	$0.597031\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.88854			−0.265772			−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854			−0.265772			−0.132886	+	0.991131i	$0.542424\pi$
$882$		0			0
$883$	−6.73722	−	6.73722i	−0.226725	−	0.226725i	0.584598	−	0.811323i	$-0.301252\pi$
$883$	−6.73722	−	6.73722i	−0.226725	−	0.226725i	−0.811323	+	0.584598i	$0.801252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			8.32766i			0.279615i	0.990179	+	0.139808i	$0.0446484\pi$
$887$			8.32766i			0.279615i	−0.990179	+	0.139808i	$0.955352\pi$
$888$		0			0
$889$			19.0557i			0.639109i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	37.9473			1.26844
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.5579	+	16.5579i	0.552237	+	0.552237i
$900$		0			0
$901$	20.3607	−	20.3607i	0.678313	−	0.678313i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	49.1935i		−	1.63525i
$906$		0			0
$907$	33.3708	−	33.3708i	1.10806	−	1.10806i	0.114655	−	0.993405i	$-0.463424\pi$
$907$	33.3708	−	33.3708i	1.10806	−	1.10806i	0.993405	−	0.114655i	$-0.0365763\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.8098			0.490672			0.245336	−	0.969438i	$-0.421102\pi$
$911$	14.8098			0.490672			0.245336	+	0.969438i	$0.421102\pi$
$912$		0			0
$913$	41.8885			1.38631
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.9443	+	20.9443i	−0.691641	+	0.691641i
$918$		0			0
$919$			2.57339i			0.0848882i	0.999099	+	0.0424441i	$0.0135144\pi$
$919$			2.57339i			0.0848882i	−0.999099	+	0.0424441i	$0.986486\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.15298	−	9.15298i	0.301274	−	0.301274i
$924$		0			0
$925$	−37.3607	−	37.3607i	−1.22841	−	1.22841i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.47214			0.0811081			0.0405541	−	0.999177i	$-0.487088\pi$
$929$	2.47214			0.0811081			0.0405541	+	0.999177i	$0.487088\pi$
$930$		0			0
$931$	18.0509	+	18.0509i	0.591595	+	0.591595i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			50.5964i			1.65468i
$936$		0			0
$937$			22.0000i			0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$			22.0000i			0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.2361	+	22.2361i	0.724875	+	0.724875i	0.969594	−	0.244719i	$-0.0786957\pi$
$941$	22.2361	+	22.2361i	0.724875	+	0.724875i	−0.244719	+	0.969594i	$0.578696\pi$
$942$		0			0
$943$	13.9845			0.455398
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.8129	+	16.8129i	0.546347	+	0.546347i	0.925382	−	0.379035i	$-0.123744\pi$
$947$	16.8129	+	16.8129i	0.546347	+	0.546347i	−0.379035	+	0.925382i	$0.623744\pi$
$948$		0			0
$949$	−2.94427	+	2.94427i	−0.0955751	+	0.0955751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	18.4721i		−	0.598371i	−0.954195	−	0.299186i	$-0.903285\pi$
$953$		−	18.4721i		−	0.598371i	0.954195	−	0.299186i	$-0.0967150\pi$
$954$		0			0
$955$	−33.1158	+	33.1158i	−1.07160	+	1.07160i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.32145			−0.139547
$960$		0			0
$961$	−15.7214			−0.507141
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0557	−	11.0557i	0.355896	−	0.355896i
$966$		0			0
$967$		−	38.3600i		−	1.23357i	−0.787130	−	0.616787i	$-0.788434\pi$
$967$		−	38.3600i		−	1.23357i	0.787130	−	0.616787i	$-0.211566\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17.2256	+	17.2256i	−0.552796	+	0.552796i	−0.927247	−	0.374451i	$-0.877831\pi$
$971$	−17.2256	+	17.2256i	−0.552796	+	0.552796i	0.374451	+	0.927247i	$0.377831\pi$
$972$		0			0
$973$	−20.9443	−	20.9443i	−0.671443	−	0.671443i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	39.4164			1.26104			0.630521	−	0.776172i	$-0.282841\pi$
$977$	39.4164			1.26104			0.630521	+	0.776172i	$0.282841\pi$
$978$		0			0
$979$	45.7649	+	45.7649i	1.46265	+	1.46265i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.98605i			0.0952402i	0.998866	+	0.0476201i	$0.0151637\pi$
$983$			2.98605i			0.0952402i	−0.998866	+	0.0476201i	$0.984836\pi$
$984$		0			0
$985$			23.1672i			0.738168i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.11146	−	6.11146i	−0.194333	−	0.194333i
$990$		0			0
$991$	14.3972			0.457341			0.228671	−	0.973504i	$-0.426562\pi$
$991$	14.3972			0.457341			0.228671	+	0.973504i	$0.426562\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.7264	+	11.7264i	0.371751	+	0.371751i
$996$		0			0
$997$	−29.4721	+	29.4721i	−0.933392	+	0.933392i	−0.997916	−	0.0645242i	$-0.979447\pi$
$997$	−29.4721	+	29.4721i	−0.933392	+	0.933392i	0.0645242	+	0.997916i	$0.479447\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bg.1153.3		8
3.2	odd	2		1536.2.j.h.1153.3	yes	8
4.3	odd	2	inner	4608.2.k.bg.1153.4		8
8.3	odd	2		4608.2.k.bf.1153.2		8
8.5	even	2		4608.2.k.bf.1153.1		8
12.11	even	2		1536.2.j.h.1153.1	yes	8
16.3	odd	4		4608.2.k.bf.3457.1		8
16.5	even	4	inner	4608.2.k.bg.3457.4		8
16.11	odd	4	inner	4608.2.k.bg.3457.3		8
16.13	even	4		4608.2.k.bf.3457.2		8
24.5	odd	2		1536.2.j.g.1153.2	yes	8
24.11	even	2		1536.2.j.g.1153.4	yes	8
32.5	even	8		9216.2.a.bj.1.3		4
32.11	odd	8		9216.2.a.bj.1.2		4
32.21	even	8		9216.2.a.bd.1.1		4
32.27	odd	8		9216.2.a.bd.1.4		4
48.5	odd	4		1536.2.j.h.385.3	yes	8
48.11	even	4		1536.2.j.h.385.1	yes	8
48.29	odd	4		1536.2.j.g.385.2	✓	8
48.35	even	4		1536.2.j.g.385.4	yes	8
96.5	odd	8		3072.2.a.k.1.1		4
96.11	even	8		3072.2.a.k.1.4		4
96.29	odd	8		3072.2.d.g.1537.6		8
96.35	even	8		3072.2.d.g.1537.1		8
96.53	odd	8		3072.2.a.q.1.3		4
96.59	even	8		3072.2.a.q.1.2		4
96.77	odd	8		3072.2.d.g.1537.4		8
96.83	even	8		3072.2.d.g.1537.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.2	✓	8	48.29	odd	4
1536.2.j.g.385.4	yes	8	48.35	even	4
1536.2.j.g.1153.2	yes	8	24.5	odd	2
1536.2.j.g.1153.4	yes	8	24.11	even	2
1536.2.j.h.385.1	yes	8	48.11	even	4
1536.2.j.h.385.3	yes	8	48.5	odd	4
1536.2.j.h.1153.1	yes	8	12.11	even	2
1536.2.j.h.1153.3	yes	8	3.2	odd	2
3072.2.a.k.1.1		4	96.5	odd	8
3072.2.a.k.1.4		4	96.11	even	8
3072.2.a.q.1.2		4	96.59	even	8
3072.2.a.q.1.3		4	96.53	odd	8
3072.2.d.g.1537.1		8	96.35	even	8
3072.2.d.g.1537.4		8	96.77	odd	8
3072.2.d.g.1537.6		8	96.29	odd	8
3072.2.d.g.1537.7		8	96.83	even	8
4608.2.k.bf.1153.1		8	8.5	even	2
4608.2.k.bf.1153.2		8	8.3	odd	2
4608.2.k.bf.3457.1		8	16.3	odd	4
4608.2.k.bf.3457.2		8	16.13	even	4
4608.2.k.bg.1153.3		8	1.1	even	1	trivial
4608.2.k.bg.1153.4		8	4.3	odd	2	inner
4608.2.k.bg.3457.3		8	16.11	odd	4	inner
4608.2.k.bg.3457.4		8	16.5	even	4	inner
9216.2.a.bd.1.1		4	32.21	even	8
9216.2.a.bd.1.4		4	32.27	odd	8
9216.2.a.bj.1.2		4	32.11	odd	8
9216.2.a.bj.1.3		4	32.5	even	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.23607 - 2.23607i) q^{5} -1.74806i q^{7} +O(q^{10})\)	\(q+(2.23607 - 2.23607i) q^{5} -1.74806i q^{7} +(-4.57649 + 4.57649i) q^{11} +(1.00000 + 1.00000i) q^{13} +2.47214 q^{17} +(4.57649 + 4.57649i) q^{19} -5.65685i q^{23} -5.00000i q^{25} +(-4.23607 - 4.23607i) q^{29} -3.90879 q^{31} +(-3.90879 - 3.90879i) q^{35} +(7.47214 - 7.47214i) q^{37} +2.47214i q^{41} +(1.08036 - 1.08036i) q^{43} +3.94427 q^{49} +(8.23607 - 8.23607i) q^{53} +20.4667i q^{55} +(6.32456 - 6.32456i) q^{59} +(-1.47214 - 1.47214i) q^{61} +4.47214 q^{65} +(8.48528 + 8.48528i) q^{67} -9.15298i q^{71} +2.94427i q^{73} +(8.00000 + 8.00000i) q^{77} +7.40492 q^{79} +(-4.57649 - 4.57649i) q^{83} +(5.52786 - 5.52786i) q^{85} -10.0000i q^{89} +(1.74806 - 1.74806i) q^{91} +20.4667 q^{95} -12.9443 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} - 16 q^{17} - 16 q^{29} + 24 q^{37} - 40 q^{49} + 48 q^{53} + 24 q^{61} + 64 q^{77} + 80 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 16 * q^17 - 16 * q^29 + 24 * q^37 - 40 * q^49 + 48 * q^53 + 24 * q^61 + 64 * q^77 + 80 * q^85 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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